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86
Affine Matching With Bounded Sensor Error: A Study of Geometric Hashing and Alignment
, 1991
"... Affine transformations of the plane have been used in a number of modelbased recognition systems, in order to approximate the effects of perspective projection. The mathematics underlying these methods is for exact data, where there is no positional uncertainty in the measurement of feature points ..."
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Cited by 50 (5 self)
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Affine transformations of the plane have been used in a number of modelbased recognition systems, in order to approximate the effects of perspective projection. The mathematics underlying these methods is for exact data, where there is no positional uncertainty in the measurement of feature points. In practice, various heuristics are used to adapt the methods to real data with uncertainty. In this paper, we provide a precise analysis of affine point matching under uncertainty. We obtain an expression for the range of affineinvariant values that are consistent with a given set of four points, where each data point lies in a disk of radius e. This analysis reveals that the range of affineinvarint values depends on the actual xypositions of the data points. That is, when there is uncertainty in the data then the representation is no longer invariant with respect to the Cartesian coordinate system. This is problematic for the geometric hashing method, because it means that the precomputed lookup table used by that method is not correct when there is positional uncertainty in the sensor data. We analyze the effect that this has on the probability that the geometric hashing method will find false positive matches of a model to an image, and contrast this with a similar analysis of the alignment method.
Finding the position and orientation of a sensor on a robot manipulator using quaternions
 International Journal of Robotics Research
, 1991
"... The problem of finding the relative position and orientation between the reference frames of a linkmounted sensor and the link has been formulated as a kinematic equation of the form Hlx = Hxcin terms of homogeneous transformation matrices by Shiu and Ahmad (1987). In this article, normalized quat ..."
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Cited by 33 (0 self)
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The problem of finding the relative position and orientation between the reference frames of a linkmounted sensor and the link has been formulated as a kinematic equation of the form Hlx = Hxcin terms of homogeneous transformation matrices by Shiu and Ahmad (1987). In this article, normalized quaternions (Euler parameters) are used to transform the kinematic equation into two simple and structured linear systems with rankdeficient coefficient matrices. Closedform solutions to these systems are derived using the generalized inverse method with singularvalue decomposition analysis. To obtain a unique solution, two distinct robot movements are required. This leads to an overdetermined system of equations. A criterion for selecting the independent set of equations is developed. A set of closedform formulae for the solution of these equations are derived. The selection criterion and the solution formulae can be easily incorporated in application programs that require the calculation of the relative position and orientation of the sensor. 1.
Solving the quintic by iteration
 Acta Math
, 1989
"... Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable ’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermit ..."
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Cited by 18 (5 self)
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Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable ’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermite and Kronecker) is replaced by a purely iterative algorithm. The algorithm requires a rational map with icosahedral symmetries; we show all rational maps with given symmetries can be described using the classical theory of invariant polynomials. 1 Introduction. According to Dickson, Euler believed every algebraic equation was solvable by radicals [2]. The quadratic formula was know to the Babylonians; solutions of cubic and quartic polynomials by radicals were given by Scipione del Ferro, Tartaglia, Cardano and Ferrari in the mid1500s. Abel’s proof of the insolvability of the general quintic
Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus
 FOUNDATIONS OF SCIENCE
, 2012
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The geometry of continued fractions and the topology of surface singularities
, 2005
"... We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a cano ..."
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Cited by 15 (1 self)
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We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers λ> 1 and λ/λ−1.
Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond
, 2012
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An Approach to Image Retrieval for Image Databases
 Lecture Notes in Computer Science
, 1993
"... In this paper, a method is discussed to store and retrieve images efficiently from an image database on the basis of the data structure called E() representation. The E() representation is a spatial knowledge representation preserving the spatial information between objects embedded in symbolic imag ..."
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In this paper, a method is discussed to store and retrieve images efficiently from an image database on the basis of the data structure called E() representation. The E() representation is a spatial knowledge representation preserving the spatial information between objects embedded in symbolic images as an iconic index for the purpose of efficient image retrieval. The image retrieval method is invariant under, at least, the affine transformation (i.e. translation, rotation and scale) and is able to deal with substantial object occlusion. A metric is defined to express similarity between symbolic images. Initial experiments carried out for two applications show that the image retrieval method is very efficient and robust to similarity retrieval in image databases. Together with the inherent high parallelism, it makes the method a promising image retrieval method. keywords: image database, image indexing, similarity retrieval, spatial relations, E() representation, metric, spatial quer...
A Probabilistic Approach to Geometric Hashing using Line Features
 Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
, 1996
"... Most current object recognition algorithms assume reliable image segmentation, which in practice is often not available. We examine the combination of the Hough Transform with a variation of Geometric Hashing as a technique for modelbased object recognition in seriously degraded single intensity im ..."
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Most current object recognition algorithms assume reliable image segmentation, which in practice is often not available. We examine the combination of the Hough Transform with a variation of Geometric Hashing as a technique for modelbased object recognition in seriously degraded single intensity images. Prior work on the performance analysis of geometric hashing has focused on point features and shown its noise sensitivity. This paper uses line features to compute recognition invariants in a potentially more robust way. We investigate the statistical behavior of these line features analytically. Various viewing transformations, which 2D (or flat 3D) objects undergo during image formation, are considered. For the case of affine transformations, which are often suitable substitutes for more general perspective transformations, we show experimentally that the technique is noise resistant and can be used in highly occluded environments. 1 1 Introduction Visual object recognition can ...
A BURGESSIAN CRITIQUE OF NOMINALISTIC TENDENCIES IN CONTEMPORARY MATHEMATICS AND ITS HISTORIOGRAPHY
, 2012
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Mathematics, mathematicians, and mathematics education
 Bulletin Amer. Math. Society
, 2005
"... I am one of a growing number of research mathematicians who are substantially engaged with school mathematics education. Such outreach has a long and honorable tradition. In this lecture, I illustrate some of the ways that I think this can be helpful, and even essential. Upon his retirement in 1990 ..."
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I am one of a growing number of research mathematicians who are substantially engaged with school mathematics education. Such outreach has a long and honorable tradition. In this lecture, I illustrate some of the ways that I think this can be helpful, and even essential. Upon his retirement in 1990 as president of the ICMI, 1 Jean Pierre Kahane spoke perceptively of the intimate connection between mathematics and mathematics education in the following terms: • In no other living science is the part of presentation, of the transformation of disciplinary knowledge to knowledge as it is to be taught (transformation didactique) so important at a research level. • In no other discipline, however, is the distance between the taught and the new so large. • In no other science has teaching and learning such social importance. • In no other science is there such an old tradition of scientists ’ commitment to educational questions.